I was trying to come up with a population model for attrition today.

Each member of the population has a risk of death per unit of time, so that you have a distribution A(r,t) which describes the distribution of the population across the axis of risk at a certain time t.

Then dA/dt = -rA, by seperation of variables, A(r,t) = A(r,0)e^(-rt)

Suppose you wanted to find the total population at time t. You'd have to integrate A(r,t) from 0 to infinity with respect to r.

It suddenly occurred to me that this looked exactly like a Laplace transform of the risk distribution function at time 0 (albeit with some variable names swapped around).

I'm left wondering if this is purely coincidence or something mathematically profound.

## Attrition population model and Laplace transform

**Moderators:** gmalivuk, Moderators General, Prelates

### Re: Attrition population model and Laplace transform

That's a very interesting observation. I've never seen the Laplace transform derived in this context before. I kind of see the Laplace transform as a way to represent the exponential components of a signal. After all, it's basically a Fourier transform in disguise. Since the population in your model is a (uncountable) linear combination of exponentially decaying functions, it makes some sort of sense that the Laplace transform is the right tool to go from the rate domain to the time domain. It's unusual to see a Laplace transform being performed on the rate variable, though.

Thanks for the cool new way to intuit the Laplace transform!

Thanks for the cool new way to intuit the Laplace transform!

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